Full Stokes to coupled ODE model

In the previous section, mobilities for the deformation and translation modes of the droplet were obtained from the full simulations. The one missing component for a quantitative com-parison of the ode model to the full dynamical simulations is the opening of the parabola used to approximate the free energy of the interface. This is shown for a 90^◦ droplet in Fig.

5.22(a), where a coefficienta=0.49 was obtained as estimator in the equation E(l) =E0+a(l−l0)²

with l0=1.59 as shown previously andE0 irrelevant, as only the gradient of the energy landscape is of any concern in the simplified model. The obtained values are a good approxi-mation up to driving forces of approximatelyµ≈0.5, as bothl0andachange less than 5% in this range and in a range for the baselengthlof±0.3. This sets an upper bound on the max-imum defect spacing and strength, as the variation in droplet baselength should stay below it.

Instead of using the expression for the energy that was introduced previously, the energy functional without the non-dimensionalization using the same parameterization as the full simulations was employed to simplify comparison of the results. It has the form:

E=a(l−l0)²−µx+w^{Z x+l/2}

x−l/2 cos(2πky/l0)dy So the final system of ODEs to be integrated takes the form

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

Fig. 5.22: (a) Free energy of the static interface over baselength for 90^◦droplet (blue), fitted (green) to obtain a spring coefficient a=0.49 for the ODE model.

(b) Contact line velocities forw=0.2,k=10,10.25,µ=0.4 showing the changing phase shift. Mobilities equivalent tols=1.

x⁰=mx∂E

∂x =mx(−µ−2wsin(2πkl/2/l0)sin(2πkx/l0)) l⁰=ml∂E

∂l =ml(2a(l−l0) +wcos(2πkl/2/l0)cos(2πkx/l0))

with a andl0 taken from the static configuration energy forµ=0 and the mobilities ml

and mx taken from the fit of the full dynamics simulations on the homogeneous substrate introduced at the beginning of the section.

Comparing the results for homogeneous substrates from this model would be moot, as the mobilities were obtained by fitting the results of full dynamic simulations on homogeneous substrates. Therefore the dynamics on a homogeneous substrate has to match. The question is if the model reproduces the features of the dynamics on a heterogeneous substrate properly, i.e. the phase shift in the contact line motion, the depinning point and the change in the bifurcation scenario.

Fig. 5.23(a) shows the obtained average droplet velocities using the mobilities for differ-ent slip lengths. It shows average velocities that are quite comparable to the full dynamical simulation results for these system parameters presented previously and a strongly increasing range of bistability for the slip length above unity. The driving force where no more pinned solution exists agrees with the stability analysis of the energy landscape. The observed range of bistability is even larger than observed in the full dynamical simulations, indicating that the higher modes of droplet deformation still play a role.

This leads to the conclusion that this change in the bifurcation scenario can be explained based on the two basic modes of droplet deformation and translation without incorporat-ing higher modes of droplet deformation for fixed contact line position. This is surprisincorporat-ing, as it defies the original assumption that an additional degree of freedom capable of storing and gradually releasing energy would be necessary to cross the energy barriers. The non-symmetric mobilities allow for the system to bypass the saddle point at an even lower slip

length, betweenls=0.1 andls=1 in this simplified model than in the full model.

A quantitative agreement can not be achieved with this model, though, for macroscopic de-fect strengths and driving forces. This is also shown by the minimum and maximum observed baselength presented in Fig. 5.23(b). First, the fluctuation of the baselength is significantly lower for the ODE model with equivalent mobilities, leading to a more pronounced bistabil-ity. That means additional deformation modes of the free interface for a given contact line position can not be neglected completely.

Second, for the full dynamical simulations the average baselength seems to be higher due to hydrodynamic deformation. This effect can not be accounted for in a linear model, as the effect is symmetric under inversion of the direction of the driving force.

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 mu

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 driving forceµ

Fig. 5.23: (a) Droplet velocities obtained from the ode model with mobilities taken from full dynamic simulations on homogeneous substrates,k= 10,w=0.2

(b) Minimal and maximal baselength of solutions from full dynamic simulations and ODE model,k=10,w=0.2

Another aspect is also reproduced qualitatively: The system also shows the changing phase shift between the front and the back contact line depending on the relative defect spacing 1/k, as illustrated in Fig. 5.22(b). For the same strength of the heterogeneity but slightly changed periodicity, the system gets closer to a complete halt withk=10.25 where the contact lines are moving nearly synchronized. For the periodicityk=10, the system approaches the pinned state for the same driving forces.

This model will fail at higher driving forces, as the assumption of the baselength dependent interfacial energy, i.e. a being independent of µ, is increasingly invalid for higher driving forces. This could be fixed by introducing a driving force dependent spring constanta, but goes beyond the scope of this strongly simplified model.

Another point that is not captured is the change of the baselength at higher velocities due to hydrodynamic deformation of the fluid interface. It can not be captured easily in a linear model, as change is independent of the direction of driving. This could be solved with a velocity-dependent term of the equilibrium baselength, though.

To conclude, for very low velocities, where the response of the system is purely linear, the ode model for the contact line motion gives a good agreement. As the higher modes of droplet deformation without displacement of the contact line and nonlinear responses of the system,

as the change in baselength due to driving, become relevant, it does not reproduce the results quantitatively.

5.5 Summary

The preceding chapter presented results on the statics and dynamics of droplets on a chemi-cally heterogeneous substrate. For a substrate with a sinusoidally varying wetting energy, the range of stable droplet states depending on the periodicity of the pattern was presented. For periodicities that are small compared to the droplet baselength, stable solutions with the range of contact angles determined by the minimum and maximum wetting energy of the droplet were observed. As the length scale of the heterogeneity approaches the droplet size, a reduced number of stable droplet configurations was observed with a hysteresis that varies with the periodicity.

When introducing a volume force parallel to the substrate that acts on the droplet, the criti-cal driving force where no more pinned droplet solutions exist has been determined depending on the substrate properties. For a defect length scale much smaller than the droplet size, the depinning force is directly determined by the wetting energy contrast. In contrast, as the pe-riodicity approaches the length scale determined by the droplet, the pinning strength can be reduced significantly if the defect length scale is commensurable with the defect length scale.

Going beyond the statics of droplet pinning, the dynamics of driven droplets on a chemi-cally heterogeneous substrate was modeled. Depending on the periodicity of the heterogene-ity, a transition between the synchronized and desynchronized motion of the contact line was observed. For high slip lengths and a substrate commensurable with the equilibrium base-length of the droplet, a bistability was observed where both a static, pinned droplet and a periodically moving droplet are a solution. At a decreased slip length, moving droplets were only observed at driving forces where no more pinned droplet states exist.

To understand this transition, a simplified droplet model with a transition and a deformation mode was introduced. It represents the droplet motion as gradient dynamics on the energy landscape given by the static configuration energy depending on the droplet baselength and center. After estimating the mobility terms from the full droplet dynamics, the resulting set of ODEs was integrated to obtain static and periodically moving solutions. It was shown that this reduced two-dimensional model can reproduce the change in the bifurcation scenario only by varying the mobility ratio of the two modes.

The present work offered an alternative perspective on the microscopic dynamics occurring when a contact line is driven over a chemically heterogeneous substrate. The change in the macroscopic contact angle when a heterogeneity is introduced is documented and explained with a velocity-dependent microscopic contact angle due to the sampling of the chemically patterned substrate. This challenges the common assumption that the static advancing con-tact angle together with the dynamics observed on a homogeneous substrate describes the dynamics on a heterogeneous substrate.

To get a better understanding of the underlying processes, the response of a free interface to a time-periodic driving has been studied. As a result, the scaling of the fluctuation am-plitude with the distance from the contact line and the phase shift depending on the driving were obtained. These scaling differ from the results obtained using a traditional approach to model contact line dynamics on a heterogeneous substrate using a friction law, as proposed by Joanny[38]. This point should be approached experimentally to answer which model re-sembles observations more closely.

Based on these results, a simplified model for the motion over the heterogeneous substrate is proposed. The model is based on a mode expansion in terms of the displacement of the con-tact line in response to time periodic perturbations of the concon-tact angle. This makes it possible to model the contact line motion by obtaining a self-consistent solution to a set of non-linear equations instead of running the full dynamical simulations, which significantly simplifies the problem. The idea is that the contact line displacement due to the heterogeneities encountered over the course of a period should be the same as the displacement due to a time dependent variation of the contact angle of the same structure.

The model was validated by determining trajectories on substrates with a sinusoidal mod-ulation of the substrate wetting energy and on substrates with a randomized periodic modu-lation of the wetting energy. The randomized wetting patterns were created as superposition of sinusoidal variations of the wetting energy with random phase shift. On these substrates, the velocity dependent variation of the contact line displacement and time averaged wetting energies were determined. The results on the random heterogeneity present the strength of the mode expansion approach, as the numerical difficulty of the problem is determined by the smallest length scale of the underlying structure, which determines the number of modes that have to be considered. The complexity of the structure itself does not influence it.

The approach could be extended to three dimensions, i.e. to a system where the wetting energy varies along the contact line to allow for an easier comparison to experimental results.

This would require solving the thin film equation in two spatial dimensions and goes beyond the scope of this work. Discussions are underway to approach this in collaboration with another research group that implemented such a code. This would give experimentalists[26]

a tool to fit their obtained contact angle-velocity diagrams with a more suitable model than those obtained for a homogeneous substrate.

Even though this could not be achieved yet, this work clarifies that there are processes occurring between the molecular length scale and the length scale on which the system is

observed due to the non-constant wetting energy. These effects should not be neglected, but are not accounted for in current models. They go beyond the effects taken into considera-tion in the two most prominent theories for contact line moconsidera-tion, the molecular kinetic theory (MKT)[11] and the Cox-Voinov scaling [90], as MKT assumes hopping on molecular length scales and Cox-Voinov assumes a fixed microscopic contact angle in the continuum limit.

Neglecting them, as done when assuming the system goes directly from one of these limits to the other one, is not appropriate for most system observed in reality.

One limitation of the numerical results in the present work is the small separation between the length scales of the slip, the characteristic length scale of the contact line fluctuations and the system size. This problem is twofold: On one side, an increased separation requires an increased refinement of the free interface with more colocation points. On the other side, the smallest length scale also limits the maximal time step when modeling the evolution of the system, which directly effects the running time. Improvements in computational power and the code base will make it possible to obtain stronger predictions concerning the problems discussed in the present work.

In the second part, a droplet driven with constant force instead of constant velocity was considered. Looking at the statics underlying the wetting of heterogeneous substrates, similar to the works of Kalliadasis [65], it became clear how not only in three dimensions, but also in two dimensions, the minimum and maximum wetting energy do not have to represent the minimum and maximum contact angle of stable solutions. This is a result of the finite length scale of the wetting heterogeneities. It also showed how with increasing driving force the range of observable advancing and receding contact angles decreases, until one stable solution remains.

Beyond the statics of wetting of heterogeneous substrate, the dynamics of droplets moving over heterogeneous substrates with defect length scales below the droplet size was studied.

For high contact angle droplets in the high slip limit, range of bistability was observed where both pinned and depinned droplet solutions coexist for the same system parameters. It was surprising, as this is often associated with inertial effects, which do not play a role in the Stokes limit. The effect could be shown to correspond to a change in the bifurcation scenario from a SNIPER bifurcation to a homoclinic bifurcation with the corresponding change in the scaling of the velocity. The initial assumption was that this change is due to the finite relaxation time of the free interface leading to an alternative energy storage mechanism. This turned out not to be the case, as the change in the length of the free interfacial energy was observed to be very low.

The same change could be observed in a simplified ODE model for the droplet dynamics derived as gradient dynamics on the free energy presented for the statics. In this model the slip dependent dynamics of the droplet is represented by different mobilities of the transla-tion and the deformatransla-tion mode, without requiring an additransla-tional parameter to represent addi-tional energy stored in the interface. This model is also relevant to the dynamical systems community, as it motivates and enables further studies of the transition between bifurcation scenarios. While bifurcations are a recurring topic in a range of fields, the transitions between bifurcation scenarios in dynamical systems have not been characterized systematically. Even though this model reproduced the transition, it did not succeed to reproduce the fluctuation of the droplet baselength after depinning accurately. This indicates that a model incorporat-ing additional deformation modes of the free interface would be in order. It was discussed that this could be possible by deriving additional equations of motion for these building on the principle of minimum energy dissipation, as already proposed for moving contact line

hydrodynamics[57].

In this context, the question of comparability of the different droplet models, ranging from the quasi-static approximation over the ODE models, proposed in the present work and by Savva[66], to the thin film limit and the full Stokes model arises. This question was ap-proached in the present thesis, but limited by the bounds on the slip length of the BEM code.

As this question was not resolved satisfactorily, further work on this topic is necessary to determine the limits of validity of the different approximations. Understanding up to which point the models for the fluid dynamic aspect of the problem hold is important when moving to complex fluid flows. For example when coupling free interface flows with surfactant flows or colloids, the complexity of the model has to be reduced to obtain an addressable problem.

Another very relevant point is the comparability of these results to experiments. For the droplet depinning on striped substrates, a linear scaling of the velocity with the excess driving force was reported[89]. A linear scaling agrees with the predicted scaling far away from the critical point, but not close to the depinning. This indicates that more precise experiments on well-controlled substrates close to the depinning transition are indispensable. Preparing sub-strates with a controlled heterogeneity is very difficult, just as observing the dynamics close to the contact line in an evolving system. But recent progress with confocal microscopy[1]

and micro-fabrication techniques should allow for this problem to be revisited experimentally.

In addition, a code to study free interface flow problems in the Stokes limit using boundary element methods was implemented. Beyond studying droplets on homogeneous and het-erogeneous substrates, it is suitable to study a range of problems with mixed solid and free boundary conditions. For systems where it is not necessary to resolve a contact line region over 3 or more orders of magnitude, for example when particles are suspended in the fluid, the performance should be quite satisfactory. Possible applications range from modeling foam formation from individual gas bubbles over suspension dynamics to active swimmers.

Another field where this code could be applied is the problem of electrowetting applied to heterogeneous substrates. Having a tunable additional energy source in the system that en-ables the droplet to cross local energy barriers makes it possible to probe the substrate energy landscape in detail.

Symbol Description

G(x,x0) Green function of a free Stokes flow T(x,x0) stress Green function of a free Stokes flow u(x) velocity of the fluid flow

σ(x) stress tensor of the fluid flow

f(x) projection of the stress tensor on the interface normal vector W(h) interface potential

η viscosity of the fluid, non-dimensionalized to unity γ surface tension, non-dimensionalized to unity

θ contact angle

k spatial periodicity of the substrate heterogeneity w0 average wetting energy of the substrate

∆w amplitude of the wetting heterogeneity ω frequency of the contact angle variation µ,g strength of the driving force

l0 equilibrium baselength of the droplet for wetting energyW0

xf position of the front contact line xb position of the back contact line l(t) baselength of the droplet over time

xc(t) coordinate of the center of the droplet base over time

`(t) length of the free interface of the droplet over time u velocity of the droplet

a spring coefficient used to model the energy of the free interface mx mobility of translation mode in center-of-base direction

ml mobility of deformation mode in baselength direction β ratio of center-of-base mobility and deformation mobility

2.1 Illustration of a pressure driven channel flow with finite slip length, with the blue parabola showing the extrapolated flow field reaching zero outside of the channel . . . 11 2.2 Three-phase contact line with the interfacial free energies of the fluid-liquid

2.1 Illustration of a pressure driven channel flow with finite slip length, with the blue parabola showing the extrapolated flow field reaching zero outside of the channel . . . 11 2.2 Three-phase contact line with the interfacial free energies of the fluid-liquid

Im Dokument Contact Line Dynamics on Heterogeneous Substrates (Seite 96-118)